Proof:$\Rightarrow$ Suppose that $p$ is prime and assume that $p \geq 5$ (the result can be checked for $p = 2$ and $p = 3$ trivially). Then we can separate the numbers $2$, $3$, …, $p - 2$ into $\frac{p - 3}{2}$ pairs of numbers $a$ and their inverses $a'$ modulo $p$ to get: